Vegetarians don’t eat meat (including fish, although some religious sects have a marketing deal with fishermen to count fish as a vegetable for some reason), do eat animal products.
Vegans don’t eat meat and also don’t eat animal products.
I always found this a funny etymological thing. Going just by the words themselves, if you asked me to guess which was which, I'd say they should be reversed -- "vegetarian" seems to clearly imply "eats a diet made up of vegetables" whereas "vegan" is more ambiguous and seems to fit "mostly eats vegetables but also will eat some non-meat animal products".
But what I'm guessing happened is that the less-strict diet came first (at least in the modern era), so got the "vegetarian" moniker, and then we just needed another work for the strict no-animal-products diet so had to come up with "vegan".
The term goes back to the Vegetarian Society in England, and was also popularized by Seventh Day Adventists in the US. The SDA church is fairly large at about 22 million members, and they’ve always promoted vegetarianism, meaning meat free diet with animal products.
Vegetarians don’t eat meat (including fish, although some religious sects have a marketing deal with fishermen to count fish as a vegetable for some reason), do eat animal products.
Vegans don’t eat meat and also don’t eat animal products.